Spatial coherence of backscatter for the nonlinearly produced second harmonic for specific transmit apodizations

To be successful, correlation-based, phase-aberration correction requires a high correlation among backscattered signals. For harmonic imaging, the spatial coherence of backscatter for the second harmonic component is different than the spatial coherence of backscatter for the fundamental component. The purpose of this work was to determine the effect of changing the transmit apodization on the spatial coherence of backscatter for the nonlinearly generated second harmonic. Our approach was to determine the effective apodizations for the fundamental and second harmonic using both experimental measurements and simulations. Two-dimensional measurements of the transverse cross sections of the finite-amplitude ultrasonic fields generated by rectangular and circular apertures were acquired with a hydrophone. Three different one-dimensional transmit apodization functions were investigated: uniform, Riesz, and trapezoidal. An effective apodization was obtained for each transmit apodization by backpropagating the values measured from within the transmit focal zone using a linear angular spectrum approach. Predictions of the spatial coherence of backscatter were obtained using the pulse-echo Van Cittert-Zernike theorem. In all cases the effective apodization at 2f was narrower than the transmit apodization. We demonstrate that certain transmit apodizations result in a greater spatial coherence of backscatter at the second harmonic than at the fundamental.     

Impact of propagation through an aberrating medium on the linear effective apodization of a nonlinearly generated second harmonic field

Techniques based on the nonlinearly generated second harmonic signal (tissue harmonic imaging) have rapidly supplanted linear (fundamental) imaging methods as the standard in two-dimensional echocardiography. Enhancements to the compactness of the nonlinearly generated second harmonic (2f) field component with respect to the fundamental (1f) field component are widely considered to be among the factors contributing to the observed image quality improvements. The objective of this study was to measure the impact of phase and amplitude aberrations resulting from propagation through an inhomogeneous tissue, on the beamwidths associated with: the fundamental (1f); the nonlinearly generated second harmonic (2f); and the linearly propagated, effective apodization signal at the same (21) frequency. Modifications to the transmit characteristics of a phased-array imaging system were validated with hydrophone measurements. Results demonstrate that the characteristics of the diffraction pattern associated with the linear-propagation effective apodization transmit case were found to be in good agreement with the detailed spatial characteristics of the nonlinearly generated second harmonic field. The effects of the abdominal wall tissue aberrators are apparent for all three of the beam profiles studied. Consistent with the improved image quality associated with harmonic imaging, the aberrated nonlinearly generated second harmonic beam was shown to remain more compact than the corresponding aberrated fundamental beam patterns in the presence of the interposed aberrator.     

Spatial coherence of synchrotron radiation

The spatial coherence properties of a monochromatic component of synchrotron radiation from an insertion device in the Fraunhofer limit are analyzed in the general case when the coherence distance is comparable with the beam width, expressing them by simple products and convolutions of Fourier transforms and autocorrelations on the single-electron field amplitude and the electron-beam position and angular distributions. In particular, the Gaussian approximation is discussed, in which case the far-field amplitude satisfies the Schell condition (its statistical properties can be described by a coherence factor depending only on the difference of the reciprocal space coordinates), and this discussion leads to simple estimates of the coherence widths. The coherence widths deviate from the Van Cittert-Zernike values when one or more of the phase space widths of the electron beam are close to (or smaller than) the diffraction limit.     

Analyzing spatial coherence using a single mobile field sensor

According to the Van Cittert-Zernike theorem, the intensity distribution of a spatially incoherent source and the mutual coherence function of the light impinging on two wave sensors are related. It is the comparable relationship using a single mobile sensor moving at a certain velocity relative to the source that is calculated in this paper. The auto-corelation function of the electric field at the sensor contains information about the intensity distribution. This expression could be employed in aperture synthesis.     

About the application of the van Cittert-Zernike theorem inultrasonic imaging

The specific circumstances under which the van Cittert-Zernike theorem applies in ultrasonic imaging systems are examined through analysis and computations. Expressions are obtained for the mutual coherence function of an incoherent source when the signals are discrete in time and space and have finite lengths. Expressions are also obtained for statistics and effective signal-to-noise ratios that describe the error in the assumption of an incoherent source with finite signal lengths. Images of a one-dimensional source are reconstructed for different signal lengths and different pulse windows. The results show that ultrasonic signals with a relatively long effective length are needed to satisfy the incoherence requirement for image reconstruction based on the van Cittert-Zernike theorem. Consequently, although the van Cittert-Zernike theorem may be used to estimate the coherence length of ultrasonic signals in the aperture of an imaging system, special data acquisition techniques are needed for satisfactory reconstruction of ultrasonic images when depth resolution like that in current b-scans is required     

Tissue harmonic image analysis based on spatial covariance

The van Cittert-Zernike theorem has been widely used to describe spatial covariance of the pressure field backscattered from a speckle object. Spatial covariance contains important information in the context of correlation-based correction of sound velocity inhomogeneities. Previous work was primarily based on spatial covariance analysis for linear imaging. In this paper, we extend the analysis to tissue harmonic imaging. Specifically, we investigate effects of the signal-to-noise ratio (SNR) and sound velocity inhomogeneities on spatial covariance. Results from tissue harmonic imaging are also compared with those from linear imaging. Both simulations and experiments are performed. At high SNRs, although both linear imaging and tissue harmonic imaging have spatial covariance functions close to theory, the spatial covariance of tissue harmonic imaging is consistently lower than that of linear imaging regardless of the presence of sound velocity inhomogeneities. At low SNRs, on the other hand, spatial covariance of tissue harmonic imaging is significantly affected. Because the tissue harmonic signal is much weaker than the linear counterpart, the low SNR reduces the accuracy of correlation-based estimation. It is concluded that the linear signal is more suitable for correlation-based correction of sound velocity inhomogeneities, despite the fact that tissue harmonic imaging generally has improved image quality over linear imaging     

General approach for representing and propagating partially coherent terahertz fields with application to Gabor basis sets

We discuss a general theoretical framework for representing and propagating fully coherent, fully incoherent, and the intermediate regime of partially coherent submillimeter-wave fields by means of general sampled basis functions, which may have any degree of completeness. Partially coherent fields arise when finite-throughput systems induce coherence on incoherent fields. This powerful extension to traditional modal analysis methods by using undercomplete Gaussian-Hermite modes can be employed to analyze and optimize such Gaussian quasi-optical techniques. We focus on one particular basis set, the Gabor basis, which consists of overlapping translated and modulated Gaussian beams. We present high-accuracy numerical results from field reconstructions and propagations. In particular, we perform one-dimensional analyses illustrating the Van Cittert-Zernike theorem and then extend our simulations to two dimensions, including simple models of horn and bolometer arrays. Our methods and results are of practical importance as a method for analyzing terahertz fields, which are often partially coherent and diffraction limited so that ray tracing is inaccurate and physical optics computationally prohibitive.     

Coherence theory of electromagnetic wave propagation through stratified N-layer media

The theory of second-order coherence in connection with wave propagation through a stratified N-layer (SNL) medium is developed. Especially, the influence of the SNL medium on the propagation of the coherence generated by a given state of coherence at the entrance plane of the medium is considered. The generalization of the van Cittert-Zernike theorem is obtained, and the propagation of the second-order coherence from a quasi-homogeneous surface distribution or a rough surface is calculated. Furthermore, the influence of SNL media on the coherence properties of a pulse is calculated.     

Transmitted and reflected scattering matrices from an English oak leaf

The complete Mueller matrix for an English oak (Quercus robur) leaf for a fixed azimuth angle (90 degrees) was determined immediately after plucking and a day following exposure to normal room temperature and pressures. The Mueller matrices were determined for transmitted light at observation angles ranging from 0 degrees to 24 degrees and for reflected backscattering angles from 153 degrees to 170 degrees. All the measurements were taken with a He-Ne laser light source at 0.63 microm. Since positive eigenvalues were obtained for the coherence matrix, the polarimetric measurements were physically realizable. The anisotropy parameters were determined from the Jones matrices by use of the decomposition theorem. From the M33 and M44 components of the Mueller matrices, it was found that nonspherical structures within the leaf were primarily responsible for observed transmitted light scatter, and spherical structures were mostly responsible for observed backscatter. Variations in backscatter Mueller matrix elements from a fresh leaf to a second day of observation were assumed because of changes to water vapor concentration in the leaf.     

Quasi-stationary plane-wave optical pulses and the van Cittert-Zernike theorem in time

We study the properties of quasi-stationary, partially coherent, plane-wave optical pulses in the space-time and space-frequency domains. A generalized van Cittert-Zernike theorem in time is derived to describe the propagation of the coherence function of quasi-stationary pulses. The theory is applied to rectangular pulses chopped from a stationary light source, and the evolution characteristics of such pulse trains with different states of coherence are discussed and illustrated with numerical examples.     

Phase-space formulation for phase-contrast x-ray imaging

Phase-space formulation based on the Wigner distribution has been presented for analyzing phase-contrast image formation. Based on the statistical nature and affine canonical covariance of Wigner distributions in the phase space, we show that the partial coherence effects of incident x-ray wave field on image intensity are simply accounted for by a multiplication factor, which is the reduced complex degree of coherence of the incident x-ray wave field. We show especially that with the undulator sources one cannot obtain the phase-contrast intensity by summing over the contributions from all electron positions, since the van Cittert-Zernike theorem fails in general for undulators. We derive a comprehensive formula that quantifies the effects of partial spatial coherence, polychromatic spectrum, body attenuation, imaging-detector resolution, and radiation dose on phase-contrast visibility in clinical imaging. The results of our computer modeling and simulations show how the formula can provide design guidelines and optimal parameters for clinical x-ray phase-contrast imaging systems.     

Second-order Coherence Effects Due to the Superposition of Two Independent Thermal Light Beams

A photon-coincidence counting technique has been used to observe the second-order coherence in a plane receiving light from two small incoherently illuminated apertures. Measurements made for two wavelengths, 546·1 and 435·8 nm, gave results consistent with predictions from the van Cittert-Zernike theorem.     

Harmonic spatial coherence imaging: an ultrasonic imaging method based on backscatter coherence

We introduce a harmonic version of the short-lag spatial coherence (SLSC) imaging technique, called harmonic spatial coherence imaging (HSCI). The method is based on the coherence of the second-harmonic backscatter. Because the same signals that are used to construct harmonic B-mode images are also used to construct HSCI images, the benefits obtained with harmonic imaging are also obtained with HSCI. Harmonic imaging has been the primary tool for suppressing clutter in diagnostic ultrasound imaging, however secondharmonic echoes are not necessarily immune to the effects of clutter. HSCI and SLSC imaging are less sensitive to clutter because clutter has low spatial coherence. HSCI shows favorable imaging characteristics such as improved contrast-to-noise ratio (CNR), improved speckle SNR, and better delineation of borders and other structures compared with fundamental and harmonic B-mode imaging. CNRs of up to 1.9 were obtained from in vivo imaging of human cardiac tissue with HSCI, compared with 0.6, 0.9, and 1.5 in fundamental B-mode, harmonic B-mode, and SLSC imaging, respectively. In vivo experiments in human liver tissue demonstrated SNRs of up to 3.4 for HSCI compared with 1.9 for harmonic B-mode. Nonlinear simulations of a heart chamber model were consistent with the in vivo experiments.     

Theory of partial coherence for weakly periodic media

The second-order theory of partial coherence for scalar and TE or TM fields is developed for weakly periodic media, and the van Cittert-Zernike theorem of classical coherence theory is generalized for such media. The coherence properties of a wave field, generated by a quasi-homogeneous source distribution at the entrance plane of a finite weakly periodic medium, are calculated both inside such a structure and in the far field. The second-order theory of partial coherence for pulse propagation through weakly periodic media is also developed.     

Multilocalization and the van Cittert-Zernike theorem. 1. Theory

The complex degree of coherence and the resulting van Cittert-Zernike theorem are employed to analyze the exit of an arbitrary amplitude-division interferometer with two-beam interferences. Considering that the source is a periodic array of spatially incoherent slits and assuming negligible equivalent aberrations and no vignetting, an expression for the complex degree of coherence as a function of the position of an exit point is derived. Formulas for the location, fringe spacing, and fringe localization depth of the multilocalized fringes are given.     

Ray and van cittert-zernike characterization of spatial coherence

We discuss a ray and a van Cittert-Zernike characterization of spatial coherence in condensers for projection systems. We present a rule of thumb with which to estimate the modulus of the coherence function at a given point of the illuminated object and a ray-tracing methodology with which to determine this modulus. For uniform illumination of the pupil we relate the modulus of the coherence function and the pupil-filling factor. We suggest that the rms of the angular ray spread at a given object point is an appropriate metric with which to characterize local coherence properties.     

Combined Raman-elastic backscatter lidar method for the measurement of backscatter ratios

A variation of the conventional combined Raman-elastic backscatter lidar method, the 1-2-3 lidar method, is described and analyzed. This method adds a second transmitter wavelength to the conventional combined Raman-elastic backscatter lidar. This transmitter wavelength is identical to that of the Raman receiver. One can generate the transmitted beam at this wavelength by Raman shifting the laser radiation in molecular nitrogen or oxygen. Measuring a second elastic lidar signal at the Raman-shifted wavelength makes it possible to eliminate differential transmission effects that can cause systematic errors in conventional combined Raman-elastic backscatter lidar.     

Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence

This paper describes the Shannon entropy in a partially coherent imaging system with Koehler illumination. Numerical simulation shows that the entropy has a one-to-one relationship with the normalized mutual intensity given by the van Cittert-Zernike theorem. Analytical evaluation shows that the entropy is consistent with the definition of coherence and incoherence, which is also verified by numerical simulations. Additional numerical experiments confirm that the entropy depends on the source intensity distribution, polarization state of the source, object, and pupil. Therefore, the entropy quantitatively measures the degree of coherence of the partially coherent imaging system.     

Atmospheric optical communication with a Gaussian Schell beam

We consider a wireless optical communication link in which the laser source is a Gaussian Schell beam. The effects of atmospheric turbulence strength and degree of source spatial coherence on aperture averaging and average bit error rate are examined. To accomplish this, we have derived analytic expressions for the spatial covariance of irradiance fluctuations and log-intensity variance for a Gaussian beam of any degree of coherence in the weak fluctuation regime. When spatial coherence of the transmitted source beam is reduced, intensity fluctuations (scintillations) decrease, leading to a significant reduction in the bit error rate of the optical communication link. We have also identified an enhanced aperture-averaging effect that occurs in tightly focused coherent Gaussian beams and in collimated and slightly divergent partially coherent beams. The expressions derived provide a useful design tool for selecting the optimal transmitter beam size, receiver aperture size, beam spatial coherence, transmitter focusing, etc., for the anticipated atmospheric channel conditions.